Nuprl Lemma : tuple_wf
∀[L:Type List]. ∀[F:i:ℕ||L|| ⟶ L[i]]. ∀[n:{n:ℤ| n = ||L|| ∈ ℤ} ]. (tuple(n;i.F[i]) ∈ tuple-type(L))
Proof
Definitions occuring in Statement :
tuple: tuple(n;i.F[i]),
tuple-type: tuple-type(L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
tuple: tuple(n;i.F[i]),
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
or: P ∨ Q,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
assert: ↑b,
btrue: tt,
subtract: n - m,
eq_int: (i =z j),
bool: 𝔹,
unit: Unit,
bnot: ¬bb,
compose: f o g,
nequal: a ≠ b ∈ T
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list_wf,
list-cases,
length_of_nil_lemma,
stuck-spread,
base_wf,
tupletype_nil_lemma,
map_nil_lemma,
list_ind_nil_lemma,
it_wf,
product_subtype_list,
spread_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
decidable__lt,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_subtype_base,
decidable__equal_int,
length_of_cons_lemma,
tupletype_cons_lemma,
upto_decomp2,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
false_wf,
map_cons_lemma,
list_ind_cons_lemma,
cons_wf,
non_neg_length,
set_wf,
equal-wf-base-T,
null-map,
null-upto,
decidable__assert,
null_wf,
null_nil_lemma,
lelt_wf,
null_cons_lemma,
bool_wf,
eqtt_to_assert,
assert_of_null,
eq_int_wf,
assert_of_eq_int,
btrue_wf,
not_assert_elim,
and_wf,
btrue_neq_bfalse,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
map-map,
nil_wf,
add-subtract-cancel,
add-member-int_seg2,
subtype_rel-equal,
select-cons-tl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaFormation,
sqequalRule,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
functionEquality,
productElimination,
universeEquality,
because_Cache,
applyLambdaEquality,
applyEquality,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality,
addEquality,
imageElimination,
imageMemberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionExtensionality,
equalityElimination,
independent_pairEquality
Latex:
\mforall{}[L:Type List]. \mforall{}[F:i:\mBbbN{}||L|| {}\mrightarrow{} L[i]]. \mforall{}[n:\{n:\mBbbZ{}| n = ||L||\} ]. (tuple(n;i.F[i]) \mmember{} tuple-type(L))
Date html generated:
2017_04_17-AM-09_29_13
Last ObjectModification:
2017_02_27-PM-05_29_50
Theory : tuples
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